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Theorem imaiun1 38713
Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
Assertion
Ref Expression
imaiun1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem imaiun1
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 3411 . . . 4 (∃𝑥𝐴𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑧𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
2 vex 3386 . . . . . 6 𝑦 ∈ V
32elima3 5688 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
43rexbii 3220 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
5 eliun 4712 . . . . . . 7 (⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵)
65anbi2i 617 . . . . . 6 ((𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ (𝑧𝐶 ∧ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵))
7 r19.42v 3271 . . . . . 6 (∃𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑧𝐶 ∧ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵))
86, 7bitr4i 270 . . . . 5 ((𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
98exbii 1944 . . . 4 (∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑧𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
101, 4, 93bitr4ri 296 . . 3 (∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
112elima3 5688 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵))
12 eliun 4712 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
1310, 11, 123bitr4i 295 . 2 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ 𝑦 𝑥𝐴 (𝐵𝐶))
1413eqriv 2794 1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 385   = wceq 1653  wex 1875  wcel 2157  wrex 3088  cop 4372   ciun 4708  cima 5313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-iun 4710  df-br 4842  df-opab 4904  df-xp 5316  df-cnv 5318  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323
This theorem is referenced by:  trclimalb2  38788
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